Global function fields with many rational places over the quinary field. II

LXXXVI.3 (1998)

Global function fields

with many rational places over the quinary field. II

by

Harald Niederreiter (Wien) and Chaoping Xing (Singapore)

1. Introduction. Let q be an arbitrary prime power and K a global function field with full constant field F q , i.e., with F q algebraically closed in K. We use the notation K/F _q if we want to emphasize the fact that F _q is the full constant field of K. By a rational place of K we mean a place of K of degree 1. We write g(K) for the genus of K and N (K) for the number of rational places of K. For fixed g ≥ 0 and q we put

N q (g) = max N (K),

where the maximum is extended over all global function fields K/F q with g(K) = g. Equivalently, N _q (g) is the maximum number of F _q -rational points that a smooth, projective, absolutely irreducible algebraic curve over F _q of given genus g can have. The calculation of N q (g) is a very difficult problem, so usually one has to be satisfied with bounds for N _q (g). Upper bounds for N _q (g) that improve on the classical Weil bound can be obtained by a method of Serre [15] (see also [16, Proposition V.3.4]).

Global function fields K/F _q of genus g with many rational places, that is, with N (K) reasonably close to N q (g) or to a known upper bound for N _q (g), have received a lot of attention in the literature. We refer to Garcia and Stichtenoth [1], Niederreiter and Xing [10], [11], and van der Geer and van der Vlugt [17] for recent surveys of this subject. The construction of global function fields with many rational places, or equivalently of algebraic curves over F _q with many F _q -rational points, is not only of great theoretical interest, but it is also important for applications in the theory of algebraic- geometry codes (see [13], [16]) and in recent constructions of low-discrepancy sequences (see [5], [9], [12]).

In the present paper we concentrate on the case q = 5 and extend the list of constructions of global function fields K/F ₅ with many rational places in [6, Section 5] and [8]. The motivation for this is that the recent tables of

1991 Mathematics Subject Classification: 11G09, 11G20, 11R58, 14G15, 14H05.

[277]

lower and upper bounds for N q (g) in [11] and [12] cover all genera g ≤ 50, except in the case q = 5 where they cover only the range g ≤ 22. We now close this gap by providing constructions for q = 5 and 23 ≤ g ≤ 50, and in fact for many other values of the genus. A crucial role in this is played by a general construction principle based on Hilbert class fields.

In Section 2 we review some background on Hilbert class fields and nar- row ray class extensions. Section 3 presents the general construction prin- ciple mentioned above and a list of examples for q = 5 derived from this principle. Further examples for q = 5 obtained by other methods are given in Section 4.

2. Background for the constructions. First we recall some perti- nent facts about Hilbert class fields. A convenient reference for this topic is Rosen [14]. Let F be a global function field with N (F ) ≥ 1 and distinguish a rational place ∞ of F . The Hilbert class field H _∞ of F with respect to

∞ is the maximal unramified abelian extension of F (in a fixed separable closure of F ) in which ∞ splits completely. The extension H _∞ /F is finite and its Galois group is isomorphic to the fractional ideal class group Pic(A) of the ring A of elements of F that are regular outside ∞. In the case under consideration (∞ rational), Pic(A) is isomorphic to the group Div ⁰ (F ) of divisor classes of F of degree 0. In particular, we have [H _∞ : F ] = h(F ), the divisor class number of F . For each place P of F there is an associated Galois automorphism τ _P ∈ Gal(H _∞ /F ), and the Artin symbol of P for the extension H _∞ /F is equal to τ _P . The place P corresponds to the divisor class of P − deg(P )∞ in Div ⁰ (F ). There is also a standard identification between places of F and prime ideals in A.

Next we collect some facts about narrow ray class extensions which can be found in [2, Section 7.5] and [4, Section 16]. Let F = F/F q , ∞, and A be as above and let φ be a sign-normalized Drinfeld A-module of rank 1. By [4, Section 15] we can assume that φ is defined over the Hilbert class field H _∞ , i.e., that for each z ∈ A the F _q -endomorphism φ _z is a polynomial in the Frobenius with coefficients from H _∞ . If H _∞ is a fixed algebraic closure of H ∞ and M a nonzero integral ideal in A, then we write Λ M for the A- submodule of H _∞ consisting of the M -division points. Let E _M := H _∞ (Λ _M ) be the subfield of H _∞ generated over H _∞ by all elements of Λ _M . Then E M /F is called the narrow ray class extension of F with modulus M . The field E _M is independent of the specific choice of the sign-normalized Drinfeld A-module φ of rank 1. Furthermore, E _M /F is a finite abelian extension with

Gal(E _M /F ) ' Pic _M (A) := I _M (A)/P _M (A),

where I M (A) is the group of fractional ideals of A that are prime to M and

P _M (A) is the subgroup of principal fractional ideals that are generated by

elements z ∈ F with z ≡ 1 mod M and sgn(z) = 1 (here sgn is the given sign function). We have Gal(E _M /H _∞ ) ' (A/M ) ^∗ , the group of units of the ring A/M . Thus, if Φ _q (M ) denotes the order of the latter group, then

[E M : F ] = |Pic M (A)| = h(F )Φ q (M ).

If M = Q ⁿ with a nonzero prime ideal Q in A and n ≥ 1, then Φ _q (Q ⁿ ) = (q ^d − 1)q ^d(n−1) ,

where d is the degree of the place of F corresponding to Q. Again in this situation, E M /F is unramified away from ∞ and Q. Furthermore, the de- composition group (and also the ramification group) D _∞ of ∞ in E _M /F is the subgroup D ∞ = {c + M : c ∈ F ^∗ _q } of (A/M ) ^∗ , and every place of H ∞

lying over Q is totally ramified in E M /H ∞ .

In the special case where F is the rational function field F _q (x) over F _q , the theory of narrow ray class extensions reduces to that of cyclotomic function fields as developed by Hayes [3]. In this case it is customary to take for

∞ the unique pole of x in F q (x). We will use the convention that a monic irreducible polynomial P over F _q is identified with the place of F _q (x) which is the unique zero of P , and we will denote this place also by P .

3. Examples from Hilbert class fields. We first establish a general construction principle for global function fields with many rational places that is based on Hilbert class fields.

Theorem 1. Let q be odd, let S be a subset of F q , and put n = |S|.

Choose a polynomial f ∈ F _q [x] such that deg(f ) is odd, f has no multiple roots, and f (c) = 0 for all c ∈ S. For the global function field F = F q (x, y) with y ² = f (x), assume that its divisor class number h(F ) is divisible by 2 ⁿ m for some positive integer m. Then there exists a global function field K/F q such that

g(K) = h(F )

2 ⁿ⁺¹ m (deg(f ) − 3) + 1 and N (K) ≥ (n + 1)h(F ) 2 ⁿ m , with equality if n = q.

P r o o f. Note that F is a Kummer extension of the rational function field F q (x) with

g(F ) = ¹ ₂ (deg(f ) − 1)

by [16, Example III.7.6]. For each c ∈ S the place x − c of F q (x) is totally ramified in F/F _q (x), and so is the pole of x in F _q (x). Let ∞ denote the unique place of F lying over the pole of x in F _q (x). For the principal divisor (x − c) of F we thus have

(x − c) = 2P _c − 2∞,

where all P c , c ∈ S, are rational places of F . Consequently, the divisor class of P _c − ∞ has order 1 or 2 in the group Div ⁰ (F ), and so the subgroup J of Div ⁰ (F ) generated by the divisor classes of all P _c − ∞, c ∈ S, has order dividing 2 ⁿ . It follows that there exists a subgroup of G of Div ⁰ (F ) with

|G| = 2 ⁿ m and G ⊇ J. Let H _∞ be the Hilbert class field of F with respect to the rational place ∞ and let K be the subfield of the extension H _∞ /F fixed by G, viewed as a subgroup of Gal(H ∞ /F ). Then

[K : F ] = h(F ) 2 ⁿ m .

By construction, the places ∞ and P c , c ∈ S, split completely in the exten- sion K/F , and this yields the desired lower bound for N (K). Furthermore, K/F is an unramified extension, and so the formula for g(K) follows imme- diately from the Hurwitz genus formula.

Remark. It is obvious that there is an analog of Theorem 1 with base fields F that are general Kummer extensions of F q (x) with arbitrary q, but Theorem 1 is of sufficient generality for our purposes.

From now on we take q = 5. In Table 1 we list examples of global function fields K/F ₅ with many rational places that are obtained from Theorem 1.

The table contains the following data: the value of the genus g(K), the value or a lower bound for the number N (K) of rational places, the values of n and m, the polynomial f (x), and the value of the divisor class number h(F ) of F = F 5 (x, y) with y ² = f (x). In the cases where the exact value of N (K) is indicated, it can be obtained from Theorem 1 or by other simple arguments. The divisor class numbers h(F ) have been calculated by the standard method based on the results in [16, Section V.1] and with the help of the software package Mathematica. Table 1 contains entries for g(K) = 15, 19, and 21 that improve on earlier examples in [8].

Table 1

g(K) N (K) n m f(x) h(F )

15 = 35 4 1 x(x + 1)(x + 2)(x − 1)(x

+ x

+ x − 2) 112 19 ≥ 45 4 1 x(x + 1)(x + 2)(x − 2)(x

− 2x

+ 2x − 2) 144

21 = 50 4 1 (x

− x)(x

− x + 1) 160

23 = 55 4 1 x(x + 1)(x + 2)(x − 1)(x

+ x

− 2x + 1) 176 24 = 46 1 1 x(x

+ x

+ 2x

+ x − 2) 46

27 = 52 1 1 x(x − 1)(x

− x + 2) 52

28 = 54 5 2 (x

− x)(x

− 2x − 2)(x

− 2x − 1) 576 29 ≥ 56 3 1 x(x + 1)(x + 2)(x − 1)(x

+ x

+ x − 2) 112

30 = 58 1 1 x(x

+ x

+ 2) 58

32 = 62 1 1 x(x

+ 2x

− 2x

− 2x + 2) 62

Table 1 (cont.)

g(K) N (K) n m f(x) h(F )

35 ≥ 68 3 1 x(x + 1)(x + 2)(x

+ x

− 2x − 2) 136

37 = 72 3 1 x(x + 1)(x + 2)(x

− 2x − 1) 144

39 = 76 3 1 x(x + 1)(x + 2)(x

+ x

− 2x

+ 2x + 1) 152 40 = 65 4 3 x(x + 1)(x + 2)(x − 2)(x

+ 2x

− 2x + 1) 624

41 = 80 3 1 x(x + 1)(x + 2)(x

+ x − 1) 160

43 = 84 3 1 x(x + 1)(x + 2)(x

− 2x

− 2) 168

45 = 88 3 1 x(x + 1)(x + 2)(x

+ 2x

+ 2x + 1) 176 46 ≥ 75 4 4 x(x + 1)(x + 2)(x − 2)(x

− x

− x + 2)(x

+ x + 2) 960 47 = 92 3 1 x(x + 1)(x + 2)(x

− 2x

− x − 2) 184 49 = 96 3 1 x(x + 1)(x + 2)(x

+ x

+ 2x

+ 2) 192

52 = 102 5 1 (x

− x)(x

+ x

+ 2x + 2) 544

53 = 104 3 1 x(x + 1)(x + 2)(x

+ x + 2)(x

− x + 1) 208 55 = 108 3 1 x(x + 1)(x + 2)(x

+ x

+ 2x + 2) 216 57 = 112 3 1 x(x + 1)(x + 2)(x

− 2x

+ x + 1) 224 58 ≥ 95 4 3 x(x + 1)(x + 2)(x − 2)(x

+ 2x

+ 1) 912

61 = 120 5 1 (x

− x)(x

+ x

+ 2) 640

64 ≥ 105 4 2 x(x + 1)(x + 2)(x − 2)(x

+ x + 1)(x

− x

− 2) 672 67 = 132 3 1 x(x + 1)(x + 2)(x

+ x

+ x − 2) 264 70 ≥ 115 4 2 x(x + 1)(x + 2)(x − 2)(x

+ 2x − 1)(x

− 2x

− 1) 736

76 = 150 5 1 (x

− x)(x

+ 2) 800

85 = 140 4 1 x(x + 1)(x + 2)(x − 2)(x

− x

− 1)(x

+ x + 1) 448 91 ≥ 150 4 2 x(x + 1)(x + 2)(x − 2)(x

− x

− x + 2)(x

+ x + 2) 960 94 = 155 4 1 x(x + 1)(x + 2)(x − 2)(x

+ x

− 2x + 2) 496 97 = 160 4 1 x(x + 1)(x + 2)(x − 2)(x

+ x + 2)(x

− x

− x − 1) 512 100 = 165 4 1 x(x + 1)(x + 2)(x − 2)(x

+ x

+ 2x − 1) 528

103 ≥ 170 4 1 (x

− x)(x

+ x

+ 2x + 2) 544

109 ≥ 180 4 1 (x

− x)(x

− 2x − 2)(x

− 2x − 1) 576 118 = 195 4 1 x(x + 1)(x + 2)(x − 2)(x

+ 2x

− 2x + 1) 624

121 = 200 4 1 (x

− x)(x

+ x

+ 2) 640

127 = 210 4 1 x(x + 1)(x + 2)(x − 2)(x

+ x + 1)(x

− x

− 2) 672 139 = 230 4 1 x(x + 1)(x + 2)(x − 2)(x

+ 2x − 1)(x

− 2x

− 1) 736

151 = 250 4 1 (x

− x)(x

+ 2) 800

172 = 285 4 1 x(x + 1)(x + 2)(x − 2)(x

+ 2x

+ 1) 912

181 = 300 4 1 x(x + 1)(x + 2)(x − 2)(x

− x

− x + 2)(x

+ x + 2) 960

199 = 330 4 1 x(x + 1)(x + 2)(x − 2)(x

+ x

− x − 2) 1056

4. Further examples. In this section we construct examples of global function fields K/F ₅ with many rational places that are obtained by princi- ples other than Theorem 1. In particular, we close all gaps in Table 1 in the range 23 ≤ g ≤ 50. We summarize all our examples from [6], [8], and the present paper in Table 2. We list the value g of the genus, a lower bound N for N ₅ (g), and a reference to either [6], [8], Table 1 of the present paper (abbreviated “Tb. 1”), or one of the following examples (“Ex.n” stands for Example n).

Table 2

g 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16

N 10 12 16 18 20 21 22 22 26 27 32 30 36 39 35 40 Ref [6] [6] [6] [6] [6] [6] [8] [6] [8] [8] [8] [6] [8] [8] Tb.1 [8]

g 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32

N 42 32 45 30 50 51 55 46 52 45 52 54 56 58 72 62

Ref [8] [8] Tb.1 [8] Tb.1 [8] Tb.1 Tb.1 Ex.1 Ex.2 Tb.1 Tb.1 Tb.1 Tb.1 Ex.3 Tb.1

g 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48

N 64 76 68 64 72 78 76 65 80 60 84 60 88 75 92 82

Ref Ex.4 Ex.5 Tb.1 Ex.6 Tb.1 Ex.7 Tb.1 Tb.1 Tb.1 Ex.8 Tb.1 Ex.9 Tb.1 Tb.1 Tb.1 Ex.10

g 49 50 51 52 53 55 56 57 58 61 64 67 70 76 85 91

N 96 70 104 102 104 108 101 112 95 120 105 132 115 150 140 150 Ref Tb.1 Ex.11 Ex.12 Tb.1 Tb.1 Tb.1 Ex.13 Tb.1 Tb.1 Tb.1 Tb.1 Tb.1 Tb.1 Tb.1 Tb.1 Tb.1

g 94 97 100 103 109 118 121 127 139 151 172 181 199 N 155 160 165 170 180 195 200 210 230 250 285 300 330 Ref Tb.1 Tb.1 Tb.1 Tb.1 Tb.1 Tb.1 Tb.1 Tb.1 Tb.1 Tb.1 Tb.1 Tb.1 Tb.1

Example 1. g(K) = 25, N (K) ≥ 52. Consider the function field F = F 5 (x, y) with

y ² = x(x − 1)(x − 2).

Then g(F ) = 1, h(F ) = 8, and the place x ² − 2x − 2 is inert in F/F 5 (x).

Let Q be the unique place of F lying over x ² − 2x − 2. Then deg(Q) = 4.

We distinguish the rational place ∞ of F which is the unique pole of x, and we denote by A the ring of elements of F that are regular outside ∞. Let E _Q /F be the narrow ray class extension of F with modulus Q. Then

[E _Q : F ] = |Pic _Q (A)| = h(F )Φ ₅ (Q) = 8 · 624.

For c = 0, 1, 2 ∈ F 5 we have the principal divisors (x − c) = 2P c − 2∞

in F . Let J be the subgroup of Pic _Q (A) generated by the residue classes of P ₀ , P ₁ , P ₂ modulo P _Q (A). Since P _c ² = (x−c)A for c = 0, 1, 2 and the residue class of x modulo x ² − 2x − 2 generates the group (F 5 [x]/(x ² − 2x − 2)) ^∗ of order 24, the order of J divides 24·8 = 192. Let G be a subgroup of Pic _Q (A) with |G| = 384 and G ⊇ J. Now let K be the subfield of E _Q /F fixed by G.

Then

[K : F ] = 8 · 624 384 = 13.

By considering the Artin symbols, we see that P 0 , P 1 , P 2 split completely in K/F , and ∞ also splits completely in K/F , hence N (K) ≥ 52. The only ramified place in K/F is Q, and it is totally and tamely ramified. Thus, the Hurwitz genus formula yields 2g(K) − 2 = (13 − 1) · 4, that is, g(K) = 25.

Example 2. g(K) = 26, N (K) ≥ 45. Consider the function field F = F ₅ (x, y) with

y ² = x ⁵ − x + 1.

The place x ⁵ −x+1 is totally ramified in F/F ₅ (x). Let Q be the unique place of F lying over x ⁵ − x + 1. Then deg(Q) = 5. We distinguish the rational place ∞ of F which is the unique pole of x, and we denote by A the ring of elements of F that are regular outside ∞. Let E _M /F be the narrow ray class extension of F with modulus M = Q ² . Then the 5-rank of the group Pic _M (A) ' Gal(E _M /F ) is at least 5 by the proof of [7, Theorem 3]. For c ∈ F 5 we have the principal divisors (x − c) = P c + P _c ⁰ − 2∞ in F , with different rational places P _c and P _c ⁰ . The subgroup J of Pic _M (A) generated by the residue classes of P ₀ , P ₁ , P ₂ , P ₃ modulo P _M (A) has 5-rank at most 4.

Thus, there exists a subgroup G of Pic M (A) with [Pic M (A) : G] = 5 and G ⊇ J.

Now let K be the subfield of E _M /F fixed by G. Then [K : F ] = 5. Since for each c ∈ F ₅ we have P _c P _c ⁰ = (x − c)A and

(x − c) ⁵

⁻¹ ≡ 1 mod M,

we see that G contains also the residue classes of P ₀ ⁰ , P ₁ ⁰ , P ₂ ⁰ , P ₃ ⁰ modulo P M (A). Therefore the places P 0 , P ₀ ⁰ , P 1 , P ₁ ⁰ , P 2 , P ₂ ⁰ , P 3 , P ₃ ⁰ , and ∞ split com- pletely in K/F , hence N (K) ≥ 45. The only ramified place in K/F is Q, and it is totally ramified. By [11, Theorem 1 and Lemma 3] the different exponent of Q in K/F is 8. Using also g(F ) = 2, we conclude from the Hurwitz genus formula that 2g(K) − 2 = 5 · (4 − 2) + 8 · 5, that is, g(K) = 26.

Example 3. g(K) = 31, N (K) = 72. Let L/F ₅ be the function field in

[6, Example 5.4] with g(L) = 4 and N (L) = 18. Then [L : F 5 (x)] = 9 and

all rational places of L lie over the zero of x or the pole of x in F ₅ (x). The

only ramified places in L/F 5 (x) are those lying over x ² + 2 or x ² − 2, each

with ramification index 3.

Now let K = L(y) with

y ⁴ = (x ² + 2)(x ² − 2).

Then all rational places of L split completely in the Kummer extension K/L, and so N (K) = 72. The only ramified places in K/L are those lying over x ² + 2 or x ² − 2, and g(K) = 31 follows from the genus formula for Kummer extensions (see [16, Corollary III.7.4]).

Example 4. g(K) = 33, N (K) = 64, K = F ₅ (x, y ₁ , y ₂ ) with y ⁴ ₁ = 2 − x ⁴ , y ₂ ⁴ = 2(x ⁴ + 2).

The places x − 1, x − 2, x + 1, and x + 2 split completely in K/F 5 (x), thus N (K) = 64. The field L = F ₅ (x, y ₁ ) is as in [6, Example 5.3], so g(L) = 3.

The only ramified places in the Kummer extension K/L are those lying over x ⁴ + 2, and g(K) = 33 follows from the genus formula for Kummer extensions.

Example 5. g(K) = 34, N (K) = 76. Consider the cyclotomic function field E M with modulus M = x ⁵ ∈ F 5 [x]. With the rational places P 1 = x + 1 and P ₂ = x − 1 of F ₅ (x), let K be the subfield of the extension E _M /F ₅ (x) constructed in [19, Theorem 1] (see also [18, Th´eor`eme 1]). Then in the notation of [19, Theorem 1] we have

s = s 5 (2, 5) = dlog ₅ 5e + 

log ₅ ⁵ ₂ 

= 2,

and so [K : F ₅ (x)] = 25 and N (K) ≥ 25 · 3 + 1 = 76. To calculate g(K), we proceed as in [19] and consider

S = {f ∈ F ₅ [x] : f (x) = (x + 1) ^h (x − 1) ^2j , h, j = 0, 1, . . .}

and

S r = {f ∈ S : x ^r k (f (x) − 1)} for r = 1, 2, . . .

We have to determine the three least values of r, called i 1 < i 2 < i 3 , for which S _r is nonempty. It is trivial that S ₁ and S ₅ are nonempty. From (x + 1) ² (x − 1) ² = x ⁴ − 2x ² + 1 we conclude that S ₂ is nonempty. Put

S(5) = {f ∈ (F ₅ [x]/(x ⁵ )) ^∗ : f ∈ S},

where f is the residue class of f modulo x ⁵ . Then S(5) is generated by x + 1 and x ² − 2x + 1, and so |S(5)| ≤ 25. If we had i ₃ < 5, then |S(5)| ≥ 125 by [19, Lemma 3], a contradiction. Therefore i ₁ = 1, i ₂ = 2, i ₃ = 5. In [19, Theorem 1] we thus have j 1 = 1 and j 2 = 2, and this yields

g(K) = 1 + 1

2 · 25 · 3 − 1 2



1 + 1 + 25 − 1 4 + 1



= 34.

From N ₅ (34) ≤ 83 it follows that N (K) = 76.

Example 6. g(K) = 36, N (K) = 64, K = F 5 (x, y 1 , y 2 , y 3 ) with y ² ₁ = x(x ² − 2), y ₂ ⁵ − y ₂ = x ⁴ − 1

y ₁ − 1 , y ₃ ² = x ³ − 2x ² − x − 2.

The field L = F ₅ (x, y ₁ , y ₂ ) is as in [8, Example 4], so g(L) = 11 and N (L) = 32. All rational places of L split completely in the Kummer extension K/L, hence N (K) = 64. The only ramified places in K/L are those lying over x ³ − 2x ² − x − 2, and g(K) = 36 follows from the genus formula for Kummer extensions.

Example 7. g(K) = 38, N (K) = 78. Consider the cyclotomic func- tion field E Q with Q = x ⁴ − 2 ∈ F 5 [x]. Let G be the cyclic subgroup of (F ₅ [x]/(x ⁴ − 2)) ^∗ ' Gal(E _Q /F ₅ (x)) generated by the residue class of x mod- ulo x ⁴ − 2. Then |G| = 16. Now let K be the subfield of E Q /F 5 (x) fixed by G. Then [K : F 5 (x)] = 39. The zero of x and the pole of x in F 5 (x) split completely in K/F ₅ (x), thus N (K) ≥ 78. The only ramified place in K/F 5 (x) is Q, and it is totally and tamely ramified. Therefore the Hurwitz genus formula yields 2g(K) − 2 = 39 · (−2) + (39 − 1) · 4, that is, g(K) = 38.

From N ₅ (38) ≤ 91 it follows that N (K) = 78.

Example 8. g(K) = 42, N (K) = 60, K = F 5 (x, y 1 , y 2 ) with y ₁ ² = (x ² + 2)(x ⁴ − 2x ² − 2), y ₂ ⁵ − y ₂ = x ⁵ − x

(x ² + 2)(x ⁴ − 2x ² − 2) . The field L = F 5 (x, y 1 ) is as in [6, Example 5.2], so g(L) = 2 and N (L) = 12.

All rational places of L split completely in the Artin–Schreier extension K/L, hence N (K) = 60. The only ramified places in K/L are the unique place of L of degree 2 lying over x ² + 2 and the unique place of L of degree 4 lying over x ⁴ − 2x ² − 2, thus g(K) = 42 follows from the genus formula for Artin–Schreier extensions (see [16, Proposition III.7.8]).

Example 9. g(K) = 44, N (K) = 60, K = F ₅ (x, y ₁ , y ₂ ) with y ⁵ ₁ − y 1 = x ⁵ − x

(x ² + 2) ³ , y ² ₂ = (x ² + 2)(x ⁸ − x ⁴ − x ² − 2).

The field L = F ₅ (x, y ₁ ) is as in [6, Example 5.12A], so g(L) = 12 and N (L) = 30. All rational places of L split completely in the Kummer ex- tension K/L, hence N (K) = 60. The only ramified places in K/L are the unique place of L of degree 2 lying over x ² + 2 and the places of L lying over x ⁸ − x ⁴ − x ² − 2, thus g(K) = 44 follows from the genus formula for Kummer extensions.

Example 10. g(K) = 48, N (K) = 82, K = F ₅ (x, y ₁ , y ₂ , y ₃ ) with y ² ₁ = x(x ² − 2), y ₂ ⁵ − y 2 = x ⁴ − 1

y 1 , y ₃ ² = x ³ − 2x ² − x − 2.

The field L = F 5 (x, y 1 , y 2 ) is as in [8, Example 9], so g(L) = 17 and N (L) = 42. All rational places of L, except the unique place of L lying over x, split completely in the Kummer extension K/L, hence N (K) = 82.

The only ramified places in K/L are those lying over x ³ − 2x ² − x − 2, and g(K) = 48 follows from the genus formula for Kummer extensions.

Example 11. g(K) = 50, N (K) = 70. Let L/F ₅ be the function field in Table 1 with g(L) = 15 and N (L) = 35. By the construction in the proof of Theorem 1 we have [L : F 5 (x)] = 14, and the rational places of L lie over x, x + 1, x + 2, x − 1 or the pole of x, with each rational place of L having ramification index 2 over F ₅ (x). Now let K = L(z) with

z ² = x ³ + 2x ² − x − 1.

Then all rational places of L split completely in the Kummer extension K/L, hence N (K) = 70. The only ramified places in K/L are those lying over x ³ + 2x ² − x − 1, and g(K) = 50 follows from the genus formula for Kummer extensions.

Example 12. g(K) = 51, N (K) = 104. Let E _Q /F be the same narrow ray class extension as in Example 1 and let J be the same subgroup of Pic _Q (A) as in Example 1. Let G be a subgroup of Pic _Q (A) with |G| = 192 and G ⊇ J. Now let K be the subfield of E _Q /F fixed by G. Then [K : F ] = 26. As in Example 1 we see that the places P 0 , P 1 , P 2 , and ∞ split completely in K/F , hence N (K) ≥ 104. The only ramified place in K/F is Q, and it is totally and tamely ramified. Thus, the Hurwitz genus formula yields 2g(K) − 2 = (26 − 1) · 4, that is, g(K) = 51. From N 5 (51) ≤ 115 it follows that N (K) = 104.

Example 13. g(K) = 56, N (K) = 101. Consider the cyclotomic func- tion field E _M with modulus M = x ⁷ ∈ F ₅ [x]. With the rational places P 1 = x + 1, P 2 = x − 1, and P 3 = x + 2, let K be the subfield of the exten- sion E _M /F ₅ (x) constructed in [19, Theorem 1] (see also [18, Th´eor`eme 1]).

Then in the notation of [19, Theorem 1] we have s = s ₅ (3, 7) = dlog ₅ 7e + 

log ₅ ⁷ ₂  + 

log ₅ ⁷ ₃ 

= 4,

and so [K : F ₅ (x)] = 25 and N (K) ≥ 25 · 4 + 1 = 101. To calculate g(K), we proceed as in [19] and consider

S = {f ∈ F ₅ [x] : f (0) = 1, f (x) = (x+1) ^h (x−1) ^j (x+2) ^k , h, j, k = 0, 1, . . .}

and

S _r = {f ∈ S : x ^r k (f (x) − 1)} for r = 1, 2, . . .

We have to obtain information on the five least values of r, called i ₁ < i ₂ <

i 3 < i 4 < i 5 , for which S r is nonempty. It is trivial that S 1 and S 5 are

nonempty. From (x + 1) ² (x − 1) ² = x ⁴ − 2x ² + 1 we conclude that S ₂ is

nonempty, and from

(x + 1)(x − 1) ⁸ (x + 2) ⁴ = x ¹³ + . . . + 2x ³ + 1

we conclude that S 3 is nonempty. Therefore i 1 = 1, i 2 = 2, i 3 = 3. Put S(5) = {f ∈ (F 5 [x]/(x ⁵ )) ^∗ : f ∈ S},

where f is the residue class of f modulo x ⁵ . Then S(5) is generated by 1 + x, 1 − x, and 1 − 2x, and so |S(5)| ≤ 5 ³ . If we had i ₄ = 4, then |S(5)| = 5 ⁴ by [19, Lemma 3], a contradiction. Therefore i 4 = 5. Put

S(7) = {f ∈ (F 5 [x]/(x ⁷ )) ^∗ : f ∈ S},

where f is the residue class of f modulo x ⁷ . Then S(7) is generated by 1 + x, 1 − x, and 1 − 2x. Since S(7) is contained in the 5-Sylow subgroup of (F ₅ [x]/(x ⁷ )) ^∗ , it follows from [19, Lemma 4(ii)] that |S(7)| ≤ 5 ^s = 5 ⁴ . If we had i 5 = 6, then |S(7)| = 5 ⁵ by [19, Lemma 3], a contradiction. Therefore i ₅ ≥ 7. In [19, Theorem 1] we thus have j ₁ = 1, j ₂ = 2, j ₃ = 3, j ₄ = 5, and this yields

g(K) = 1 + 1

2 · 25 · 5 − 1 2



1 + 1 + 1 + 5 + 25 − 1 4 + 1



= 56.

From N 5 (56) ≤ 125 it follows that N (K) = 101.

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Institut f¨ ur Informationsverarbeitung Osterreichische Akademie ¨

der Wissenschaften Sonnenfelsgasse 19 A-1010 Wien, Austria

E-mail: niederreiter@oeaw.ac.at

Department of Information Systems and Computer Science The National University of Singapore Lower Kent Ridge Road Singapore 119260 E-mail: xingcp@comp.nus.edu.sg

Received on 22.6.1998 (3412)